Testing Einstein's gravity and dark energy with growth of matter perturbations: Indications for new Physics?

The growth index of matter fluctuations is computed for ten distinct accelerating cosmological models and confronted to the latest growth rate data via a two-step process. First, we implement a joint statistical analysis in order to place constraints on the free parameters of all models using solely background data. Second, using the observed growth rate of clustering from various galaxy surveys we test the performance of the current cosmological models at the perturbation level while either marginalizing over $\sigma_8$ or having it as a free parameter. As a result, we find that at a statistical level, i.e. after considering the best-fit $\chi^2$ or the value of the Akaike information criterion, most models are in very good agreement with the growth rate data and are practically indistinguishable from $\Lambda$CDM. However, when we also consider the internal consistency of the models by comparing the theoretically predicted values of $(\gamma_0, \gamma_1)$, i.e. the value of the growth index $\gamma(z)$ and its derivative today, with the best-fit ones, we find that the predictions of three out of ten dark energy models are in mild tension with the best-fit ones when $\sigma_8$ is marginalized over. When $\sigma_8$ is free we find that most models are not only in mild tension, but also predict low values for $\sigma_8$. This could be attributed to either a systematic problem with the growth-rate data or the emergence of new physics at low redshifts, with the latter possibly being related to the well-known issue of the lack of power at small scales. Finally, by utilizing mock data based on an LSST-like survey we show that with future surveys and by using the growth index parameterization, it will be possible to resolve the issue of the low $\sigma_8$ but also the tension between the fitted and theoretically predicted values of $(\gamma_0, \gamma_1)$.